6. Real-World Pythagorean Problems

Learning goals

By the end of this lesson, you should be able to:

• translate a situation into a labeled right triangle;
• distinguish travel distance from straight-line displacement; and
• carry units and rounding instructions through a multi-step solution.

Most GED® Pythagorean questions are word problems. The skill is drawing (or imagining) the right triangle and labeling the sides.

A reliable routine:

• Sketch the situation and mark the right angle.
• Label the two legs and the hypotenuse (the slanted or longest distance).
• Decide: are you finding the hypotenuse (add) or a leg (subtract)?

Common setups:

• A ladder against a wall: ladder = hypotenuse; ground distance and wall height = legs.
• The diagonal of a rectangle, room, or screen: the diagonal = hypotenuse; length and width = legs.
• A ramp: the slope = hypotenuse; the run and rise = legs.
• Distance traveled (go east, then north): the straight-line distance = hypotenuse.

Worked example: a person walks 9 km east, then 12 km north. Straight-line distance from start:
\[ \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 \text{ km}. \]

The person walked \(9+12=21\) km, but finished 15 km from the start. Those quantities answer different questions. Read the final sentence carefully.

Build a model from clue words

Worked example: a rectangular garden is 18 m long and 24 m wide. A path runs diagonally from one corner to the opposite corner.
\[ d=\sqrt{18^2+24^2}=\sqrt{324+576}=\sqrt{900}=30\text{ m}. \]
If paving costs $12 per meter, the total cost is \(30\times12=$360\). The geometry result becomes an input to the second step.

Units and scale

Convert measurements to the same unit before using the theorem. Do not square 5 feet and 24 inches together. Convert 5 feet to 60 inches, or 24 inches to 2 feet, first.

If a map uses a scale, find the map distance first and then apply the scale. For a 5 cm diagonal on a map with 1 cm representing 4 km, the real distance is \(5\times4=20\) km.

Answer discipline: report the requested quantity, attach the correct unit, and round only as directed.

Common mistake: not identifying which side is the hypotenuse before computing. Always find the right angle first.